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A Simple Encryption Algorithm 1 A Simple Encryption Algorithm 2 Joseph Keenan St. Pierre 3 Worcester Polytechnic Institute 4 Corresponding author: 5 Joseph Keenan St. Pierre 6 Email address: [email protected] 7 ABSTRACT In this paper I present a Simple Encryption Algorithm (SEAL), by which 128-bit long blocks can be quickly encrypted/decrypted. The algorithm is designed to run efficiently in software without any specialized hardware while still guaranteeing a strong degree of confidentiality. The cipher is composed entirely of simple bit-wise operations, such as the exclusive-or and circular shift, in addition to modular addition, thereby making it exceedingly easy to implement in most programming languages as well as efficient in terms of performance. 8 INTRODUCTION 9 SEAL is a symmetric key block cipher that permutes a 128-bit block against a 128-bit key. Based off of 10 statistical analysis, the cipher’s output exhibits strong avalanche effects as well as uniform distribution. 11 Furthermore, as the name suggests, SEAL’s implementation in software is lightweight, straightforward, 12 and requires minimal overhead. 13 ENCRYPTION PROCEDURE 14 Below is the encryption procedure for SEAL written in the C programming language: 15 16 /*Substitute the bytes ofa block chunk with the SEALS-Box */ 17 void S(uint32_t *block_chunk){ 18 for(int i = 0; i < 4; i++){ 19 uint8_t byte = (uint8_t)(*block_chunk >> 8*i); 20 *block_chunk = *block_chunk & ˜(0xFF << 8*i); 21 *block_chunk |= ((uint32_t)sbox[byte] << 8*i); 22 } 23 } 24 /*Encryptsa 128-bit block againsta 128-bit key using SEAL */ 25 void seal_encrypt(uint32_t *block, uint32_t *key){ 26 /*The carry chunk used for rotation and feeding*/ 27 uint32_t carry = 0; 28 29 /*Perform8 rounds of encryption */ 30 for(int i = 0; i < 8; i++){ 31 //Substitute first chunk and XOR against key and round number 32 S(&block[0]); block[0] ˆ= key[i%4] ˆ i; carry = block[0]; 33 34 //Feed carry chunk through subsequent block chunks 35 block[1] += carry; carry = block[1]; 36 carry = (carry >> 11) | (carry << 21);//Rotate carry chunk 37 38 block[2] += carry; carry = block[2]; 39 carry = (carry >> 11) | (carry << 21); 40 41 block[3] += carry; carry = block[3]; 42 carry = (carry >> 11) | (carry << 21); PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.3128v3 | CC BY 4.0 Open Access | rec: 15 Aug 2017, publ: 15 Aug 2017 43 44 //Rotate block chunks 45 block[3] = block[2]; block[2] = block[1]; 46 block[1] = block[0]; block[0] = carry; 47 } 48 49 S(&block[0]); S(&block[1]);//Substitute chunks again 50 S(&block[2]); S(&block[3]); 51 52 for(int i = 0; i < 4; i++)//Perform"locking" round 53 block[i] ˆ= key[i];//XOR chunks against key 54 } 55 56 SUBSTITUTION BOX 57 For added reference, below is the 16x16 lookup table used by the S-box to substitute the bytes of a block 58 chunk. This table was generated randomly using a brute-force algorithm for finding a substitution box 59 with strong avalanche effect. Any lookup table will suffice provided it too has good bit distributions 60 and a strong avalanche effect. A byte is substituted by taking its hexadecimal value, reading its first and 61 second digit which correspond to a row and column respectively, and replacing said byte with the value 62 located at the corresponding cell. For example, the hexadecimal value ”ab” would be substituted with 63 ”67” according to the provided table. 64 Table 1. S-box S 0 1 2 3 4 5 6 7 8 9 a b c d e f 0 bc 7d 7b 93 01 15 30 21 a5 d6 f8 9b 48 db ce 29 1 61 b3 34 03 b1 d7 53 98 52 f3 bb ab b2 2d 2a eb 2 08 02 0c cb b0 9e 8f 96 40 92 e9 6e 58 6d 44 06 3 88 1c 1f 65 91 85 66 45 9d 4a b9 8d 20 ca a0 19 4 5d 5b 12 25 cc 9c 43 a2 50 da b4 f9 4f 69 17 4b 5 04 6c 11 dd 73 16 df 41 3a 5f 74 47 09 18 fe 99 6 84 62 00 0d 64 7c d5 72 e1 e5 24 ee 4d f2 3d 2c 65 7 26 3b 42 3f c2 7a d3 1d 57 0b fa 75 d0 c4 ec fb 8 0e e3 90 80 ff 0a 4e 2f d9 2e c8 e6 1b 94 55 9a 9 f5 60 79 d2 71 cf dc ad f7 7f c0 05 6b af 38 7e a d8 35 70 c9 aa 83 a6 d1 39 e0 6a 67 a4 5a 13 8b b f0 fc e7 c6 a3 97 c3 5e c7 63 46 ba 37 ea 77 c1 c cd 4c 33 bf 28 76 b5 b7 f4 ed 5c fd 68 ae e4 78 d e2 1e 2b ac 59 36 be 6f 1a b6 9f 22 87 8c 10 31 e 82 a9 07 f6 86 23 e8 95 54 a8 82 8a a1 49 f1 b8 f d4 3e 0f de 3c 8e 56 c5 27 89 14 bd 51 a7 32 ef 66 OVERVIEW AND DESIGN RATIONALE 67 The cipher is a form of substitution-permutation network whereby both the block and the key are divided 68 into equal 32-bit chunks. Through the combined usage of modular additions, circular shifts, and the 69 XOR, a high degree of non-linearity is achieved. Prior to the start of each round, the round-key is XOR’d 70 against the round number; this is done in order to protect the cipher against slide attacks by removing 71 the symmetry of the internal rounds. The substitution boxes are included in order to rapidly introduce 72 the avalanche effect resulting from a minor change in the input block. Furthermore, to ensure that any 73 change in the key results in a major change in the output, at least eight rounds of encryption are required; 74 however, if additional security is needed, the number of rounds can be scaled up provided that the number 75 of rounds is a multiple of four. This guarantees that each chunk of the key is used the same number of 76 times thereby preventing any bias from emerging. Lastly, in order to obfuscate the internal permutations 77 of the block, the entire block is substituted and XOR’d against the key in a final ”locking” round. 2/7 PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.3128v3 | CC BY 4.0 Open Access | rec: 15 Aug 2017, publ: 15 Aug 2017 78 SEAL has been specifically designed for efficiency in software without the need for dedicated 79 hardware. For this reason, there is no key schedule in the default implementation presented here as such 80 an additional process unnecessarily slows the rate of encryption. This omission is justified on the basis 81 that the statistical analysis of the cipher’s output distribution and avalanche effect showed no conclusive 82 advantage to using SEAL with a key schedule as opposed to using the same key repeatedly. That being 83 said, if performance is not a concern, a key schedule could easily be implemented and used alongside 84 SEAL if one so desired. 85 CIPHER DIAGRAM 86 Below is an overall outline of the SEAL cipher: Figure 1. SEAL Cipher Digram 3/7 PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.3128v3 | CC BY 4.0 Open Access | rec: 15 Aug 2017, publ: 15 Aug 2017 87 ANALYSIS 88 Substitution Box 89 The primary function of an S-box is to remove the linear relationship between a plaintext-ciphertext 90 pair produced by a cryptographic function. In order to effectively accomplish this task, it is imperative 91 that an S-box does not have any strong differential characteristics that could be feasibly exploited by 92 cryptanalysis. 93 The S-box articulated in this paper was produced via a brute-force algorithm that searched for a 94 16x16 lookup table that followed an avalanche criterion in addition to possessing only weak differential 95 characteristics; as such, the best differential characteristic across the SEAL S-box, (e5 ! 8f), only holds 96 true with a probability of 0.023438. 97 While the S-boxes produced by this algorithm are probably not secure enough to resist an extensive 98 differential attack on their own, it should be noted that when used in conjunction with other non-linear 99 functions (such as modular addition), the cipher as a whole becomes infeasible to attack. 100 Internal Round Functions 101 SEAL’s internal round functions in and of themselves are not secure as only the first chunk is technically 102 kept a secret via the XOR. However, when used in a group of 4, the entire block is permuted and cannot 103 easily be recovered despite the fact that intermediary computations can be trivially found if the ciphertext 104 output is known. This is best articulated using a 1x4 matrix which represents the outputted permuted 105 block, a 4x4 matrix which represents the states of the various XOR and ADD nodes for each internal 106 round, and a final 1x4 matrix which represents the original inputted plaintext block. For demonstration 107 purposes, the matrices below will only contain 8-bit values and the circular shifts will be by 3 bits instead 108 of 11. C Round1−4 P 2ab3 2ab a0 c2 5c3 2xx3 6167 616 d6 e4 xx7 6xx7 6 7 ( 6 7 ( 6 7 4995 499 b f xx xx5 4xx5 f 2 f 2 xx xx xx xx Figure 2.
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